Mash garlic and salt together, rub some of the paste onto the top of the artichoke bottoms and place upside down in a 12″ saute pan. Add water, olive oil, the rest of the garlic paste, and lemon half. Bring to a boil, reduce to a simmer, cover and cook for 20 minutes.

Remove artichoke bottoms from the pan, sprinkle with basil. Bring the braising liquid to a boil, and reduce to 1/4 cup, whisking to emulsify. Drizzle over artichoke bottoms. Serve at room temperature.